An Overview of Sieve Methods

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 2, 67 - 80

R. A. Mollin

Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 URL: http://www.math.ucalgary.ca/˜ ramollin/ [email protected]

Abstract

We provide an overview of the power of Sieve methods in number theory meant for the non-specialist.

Mathematics Subject Classiﬁcation: Primary: 11N35; Secondary: 11- 02; 11N36

Keywords: Sieves, open problems

1 Sieves

Some of the following is adapted from [11]. Sieve methods are used in fac- toring, recognizing primes, finding natural numbers in arithmetic progression whose common difference are primes, or generally to estimate the cardinalities of various sets defined by the use of multiplicative properties. Recall that use of a sieve or sieving is a process whereby we find numbers via searching up to a prescribed bound and eliminate candidates as we proceed until only the desired solution set remains. In other words, sieve theory is designed to estimate the size of sifted sets of integers. For instance, sieves may be used to attack the following open problems, for which sieve methods have provided some advances. (a) (The Twin Prime Conjecture) There are infinitely many primes p such that p + 2 is also prime.

(b) (The Goldbach Conjecture) Every even integer n>2 is a sum of two primes. 68 R. A. Mollin

(d) (The q = 4p + 1 Conjecture) There are inﬁnitely many primes p such that q =4p + 1 is also prime.

(e) (Artin’s Conjecture) For any nonsquare integer a ∈{−1, 0, 1}, there exist inﬁnitely many primes p such that a is a primitive root modulo p.

Indeed, in 1986, Heath-Brown [8] used sieving methods to advance the Artin conjecture to within a hair of a solution when he proved that with the possible exception of at most two primes, there are inﬁnitely many primes q such that p is a primitive root modulo q. Thus, sieve methods are important to review for their practical use in number theory and the potential for solutions of outstanding problems such as the above. The fundamental goal of sieve theory is to produce upper and lower bounds for cardinalities of sets of the type, S(S, P,y)={n ∈ S : p n implies p>yfor all p ∈ P}, (1) where S is a ﬁnite subset of N, P is a subset of P, the set of all primes, and y is a positive real number.

Example 1 Let √ S = {n ∈ N : n ≤ x} and x

Then |S(S, P,y)| = {n ≤ x : p n implies p>y} = π(x) − π(y)+1, one more than the number of primes between x and y.

To illustrate (1) more generally, we begin with what has been called “the oldest nontrivial algorithm that has survived to the present day.” From antiq- uity, we have the Sieve of Eratosthenes, which is covered in a first course in number theory—see [10, Example 1.16, p. 31], which sieves to produce primes to a chosen bound. However, as discussed therein, this sieve is highly ineffi- cient. Indeed, since in order to determine the primes up to some bound using ∈ N √this sieve for n , one must check for divisibility by all primes not exceeding n, then the sieve of Eratosthenes has complexity O(n loge n)(loge loge n)), An overview of Sieve methods 69 which even using the world’s fastest computers, this is beyond hope for large integers as a method for recognizing primes. Yet there is a formulation of this sieve that fits nicely into the use of arithmetic functions, and has applications as a tool for modern sieves, so we present that here for completeness and interests sake. Recall that the Möbius function μ(d) is defined by 1ifn =1, μ(n)= 0ifn is not squarefree, k k (−1) if n = j=1 pj where the pj are distinct primes.

Also, let ω(d) denote the number of distinct prime divisors of d, and P the set of all primes.

Theorem 1 Eratosthenes’ Sieve

Let P = {p1,p2,... ,pn}⊆P be a set of distinct primes and let S ⊆ N with |S| < ∞. Denote by S the number of elements of S not divisible by any of the pj’s and by Sd the number of elements of S divisible by a given d ∈ N. Then S = μ(d)Sd.

d|p1p2···pn

Moreover, For m =1, 2,... ,n/2 , we have μ(d)Sd ≤ S ≤ μ(d)Sd, d|p p ···p d|p p ···p 1 2 n 1 2 n ω(d)≤2m−1 ω(d)≤2m where (1) is called Eratosthenes’ sieve.

Proof. See [12, Corollary 2, p. 147]. 2 For instance, an application of Theorem 1 is that it may be used to prove the following result on the number of primes less than a certain bound, ﬁrst proved in 1919, by the Norwegian mathematician Viggo Brun (1882–1978).

Theorem 2 Brun’s Theorem

If n ∈ N and B2n(x) denotes the number of primes p ≤ x for which |p +2n| is also prime, then

2 −2 B2n(x)=O(x(loge loge x) loge x). 70 R. A. Mollin

Proof. See [12, Theorem 4.3, p. 148]. 2 Theorem 2 has as a special case, implications for the twin prime conjecture as follows.

Corollary 1 Brun’s Constant Let Q be the set of all primes p such that p +2 is also prime, then

2 x(loge loge x) 1 << 2 ,

loge x p∈É p≤x and the series 1 = B (2)

p p∈É is convergent, where (2) is called Brun’s constant.

Proof. See [12, Corollary, p. 152]. 2

Remark 1 We do not know if Q in Corollary 1 is finite or not since its infinitude would be the twin prime conjecture. We do know that the sum of the reciprocals of all primes diverges, but since the series (2) converges, this is not a proof of the conjecture since we would need divergence to get the infinitude. The behaviour of the two series does tell us that, although the twin prime conjecture may be true, the twin primes must be appreciably less dense than the entire set of primes. Brun’s result, that the reciprocals of twin primes converges, is one of the centerpiece achievements of sieve theory. The value of Brun’s constant is

B ≈ 1.9021605824, with an error within ±0.000000003, computed by Thomas R. Nicely in 1999. It is worth noting the now famous fact that, in 1995, Nicely was doing computa- tions on Brun’s constant which led him to discover a ﬂaw in the ﬂoating-point arithmetic of the Pentium computer chip, costing literally millions of dollars to its manufacturer Intel—see http://www.trnicely.net/twins/twins2.html.

Theorem 1 tells us that the sieve of Eratosthenes investigates the function

S(S, P,x)= 1, where Π = p

n∈Ë p∈È gcd(n,Π)=1 p≤x An overview of Sieve methods 71 via the equality S(S, P,x)= μ(d)= μ(d)Sd. d|n n∈Ë d|Π d|Π The general basic sieve problem emanates from this, namely ﬁnd arithmetic functions λ(d):N → R and λu(d):N → R with 1 if gcd(n, Π) = 1, λ(d) ≤ 0 if gcd(n, Π) > 1, d|n d|Π and 1 if gcd(n, Π) = 1, λu(d) ≥ 0 if gcd(n, Π) > 1, d|n d|Π such that λ(d)Sd = λ(d) ≤ S(S, P,x) ≤ λu(d)= λu(d)Sd. (3)

d|n d|n n∈Ë d|Π n∈Ë d|Π d|Π d|Π

Now we interpret the above in terms of what Selberg did to create his famous sieve and how Theorem 1 comes into play. With the notation of Theorem 1 still in force, we add that P denotes the product of the primes in P, |S| = N, and call the following Selberg’s condition on S. There exists a multiplicative function f(d) such that if d P , then f(d) Sd = N + R(d), (4) d where |R(d)|≤f(d) and d>f(d) > 1. With the Selberg condition plugged into the right-hand side of (3), we have λu(d)f(d)N |S(S, P,x)|≤ + λu(d)R(d) d|Π d d|Π ⎛ ⎞ λu(d)f(d) ⎝ ⎠ = N + O |λu(d)R(d)| . (5) d|Π d d|Π

Selberg’s sieve arose from his attempts to minimize (5) subject to Selberg’s condition (4). Theorem 1 comes into play again in that it is used in the proof of the following, ﬁrst proved by Selberg [13] in 1947. The following is considered to be the fundamental theorem concerning Selberg’s sieve, which for the above-cited reasons, is often called Selberg’s upper bound sieve. 72 R. A. Mollin

Theorem 3 Selberg’s Sieve Let P be a ﬁnite set of primes, P denoting their product, S ⊆ N with |S| = N ∈ N, such that S satisﬁes Selberg’s condition (4), and let S = S(S, P,x) be the number of elements of S not divisible by primes p ∈ P with p ≤ x where x>2. If for p P , we have that f(p) > 1,

μ(n/d)d g(n)= , d|n f(d) and −1 Qx = g (d), d|P d≤x then −2 N f(p) S ≤ + x2 1 − .

Qx p p∈È p≤x

Proof. See [12, Theorem 4.4, p. 158]. 2 An application of Theorem 3 is the following, where π(x; k, ) denotes the number of primes p ≤ x such that p ≡ (mod k). In the notation of Theorem 3, we have that √ P = {p ∈ P : p k and p ≤ x}. Also, S = {y = kn + : n ∈ N and y ≤ x}. Then N = x/k , √ √ S(S, P,x)=π(x; k, ) − π( x; k, )=π(x; ,k,)+O( x).

| |≤ It follows that f(d)=1,Sd = N/d + Rd with Rd 1, g(n)=φ(n), and −1 Qx = x≥d|P φ (d).

Theorem 4 The Brun-Titchmarsh Theorem There exists a C = C(ε) ∈ R+ such that for 1 ≤ q

Proof. See [12, Corollary, p. 161]. 2 An overview of Sieve methods 73

Remark 2 Theorem 4 is known to hold when the constant c = 2. Moreover, if 1 ≤ q ≤ x1−ε for ε>0, then the upper bound is at the expected order of magnitude. Another interpretation of Theorem 4 is that if x, y are positive reals, and k, ∈ Z with y/k →∞, then

(2 + o(1))y π(x + y, k, ) − π(x, k, ) < . φ(k) loge(y/k)

Yet another formulation is given as follows. There exists an eﬀective constant k>k0(ε) such that

(2 + ε)y π(x + ky, k, ) − π(x, k, ) < , φ(k) loge y for all y, x, with y>k. The amazing aspect of Brun-Titchmarsh is that if we could replace 2 by 2 − δ for any δ>0, then Landau-Siegel zeros cannot exist.

Selberg’s sieve also has applications to some other classical problems. For instance, the twin-prime conjecture may be interpreted as follows. Suppose that f(d) represents the number of elements of {n(n +2):d n(n + 2) where 1 ≤ n ≤ d} which are divisible by d and for some m ∈ N,

S = {j(j +2):j = m, m +1,... ,m+ N − 1}.

Let π2(N) be the number of twin primes less than N, from which it follows that 1/3 1/3 π2(N) ≤|S(S, P,N )| + N because if p ≤ N has a twin prime, then either p ≤ N 1/3 or else p(p+2) has no prime factor ≤ N 1/3. Thus, using Selberg’s sieve to estimate |S(S, P,N1/3)|, we have f(2) = 1 and f(p) = 2 for odd primes p. We claim that

−1 f(p) 1 − << (log N)2. p e p≤N 1/3

This follows from the fact that for p>3,

−1 −2 −1 2 1 2 1 − ≤ 1 − 1 − p p p2 74 R. A. Mollin

and the fact that −1 1 1 − << log N 1/3, p e p≤N 1/3 which, in turn, follows from Merten’s Theorem, keeping in mind that p≤N 1/3 (1− 2p−2)−1 converges. (Recall that Merten’s theorem says: 1 = loge loge x + M + o(1), p≤x p and 1 1 M = γ + loge 1 − + , p=prime p p where γ is Euler’s constant and M is called Merten’s constant.) One may also deduce a lower bound as follows, f(d) ≥ (log N)2. d e d≤N1/3 d odd Putting this all together via Theorem 3, we get the following.

Theorem 5 Selberg’s Sieve on Twin Primes

The number π2(N) of twin primes less than N satisfies N π2(N) << 2 . (loge N) Remark 3 With the above application of Selberg’s sieve, it is certainly worth mentioning another highlight of sieve theory with respect to the twin-prime conjecture, namely Chen’s Theorem, which shows that there are infinitely many primes p such that p+2 is either prime or a product of two primes. Again, sieve methods allowed a result that is within a hair of the affirmation of another classical conjecture.

Another of the list of conjectures from our discussion at the outset is the Goldbach conjecture. Now we look at applications of Selberg’s sieve to this classical problem. To this end, let N =2m for m ∈ N, and for some k ∈ N,

S = {j(N − j):j = k, k +1,... ,k+ N − 1}, and let r(N) be the number of representations of N as a sum of two primes. Also, f(d) is the number of elements of

{n(N − n):n =1, 2,... ,d} An overview of Sieve methods 75 divisible by d. It follows that

r(N) ≤|S(S, P,N1/3)| +2N 1/3. Thus, f(p)=2ifp N and f(p)=1ifp N. Applying Theorem 3, and arguing in a similar fashion to the above, we get the following, a complete proof of which may be found in [12, Theorem 4.6, p. 162].

Theorem 6 Selberg’s Sieve on the Goldbach Conjecture For N ∈ N, r N 2 (N) << 2 1+ . (loge N) p|d p

We have amply illustrated the applications of Selberg’s sieve to a variety of classical problems. It is now time to look at other sieves and their contribu- tions. One of these is due to Linnik [9] ﬁrst produced in 1941. To understand what it says, we provide a preamble that takes into account what we have learned thus far. Brun’s result Theorem 2 may be interpreted as a general- ization√ of Eratosthenes sieve as follows. Take 1, 2,... ,n and for each prime p ≤ n, we eliminate k residue classes modulo p, then the number remain- k ing does not exceed C(k)N/(loge n), where C(k) > 0 depends on k. Linnik√ considered a more general situation by considering for each prime p ≤ n, we eliminate f(p) classes modulo p where f(p) gets large as p does. Linnik called this the large sieve. This is formalized in terms of the notation we have developed herein as follows.

Theorem 7 The Large Sieve Inequality √ Suppose that N ∈ N and for every prime p ≤ N, let f(p) residue classes modulo p be given, where 0 ≤ f(p)

(1 + π)N √ − p≤ N f(p)/(p f(p)) integers not lying in any of the given residue classes.

Proof. See [12, Corollary 2, p. 170]. 2 The large sieve can be applied to Artin’s conjecture, one of the classical problems from our list at the outset. From the large sieve Theorem 7, we have the following. 76 R. A. Mollin

Theorem 8 The Large Sieve on Artin’s Conjecture

Let IN be an interval of natural numbers of length N ∈ N and let √ C(N)= n ∈ IN : n is not a primtive root modulo for any prime p ≤ N .

Then √ C(N) << N loge(N).

Proof. See [12, Theorem 4.8, p. 171]. 2

Corollary 2 Almost every n ∈ N is a primitive root for some prime.

Using the large sieve, Bombieri [1] and Vinogradov [15] independently found a result on distribution of primes in arithmetic progression that is quite pleas- ant. In the next result, we use the following. The (basic) Mangoldt function is given by

a Λ(n) = loge p if n = p for some prime and p, a ∈ N, and Λ(n) = 0 otherwise.

In the

Theorem 9 The Bombieri-Vinogradov Theorem

For√ any real number A>0, there is a constant B = B(A) such that, for −B Q = x(loge x) , y x max max ψ(y; q,a) − << , (6)

y≤x ∗ A /q a∈( ) q≤Q φ(q) (loge x) where ψ(x; q,a)= Λ(n). n≤x n≡a (mod q)

In keeping with the above, we now show how some classical problems can be tackled with Theorem 9. If τ(x) is the number of divisors function, and n ∈ N, is ﬁxed, then the Titchmarsh divisor problem is to compute the order of the function S(x)= τ(p + n). p≤x Theorem 9 can be applied to this problem to get the following—see [12, The- orem 5.11, p. 202] for a related result. An overview of Sieve methods 77

Theorem 10 Bombieri-Vinogradov Applied to Titchmarsh For any n ∈ N, there exists a constant c ∈ R+ such that x log log x S(x)=cx + O e e . loge x

This establishes more than that proved by Titchmarsh [14] in 1930, wherein he showed that S(x)=O(x). Bombieri also provided a sieve, essentially generalizing the Selberg sieve, that was highly useful in establishing another highlight of sieve theory. To describe this and the application, we need the following notions. If (6) holds for any A>0 and any ε>0 with Q = xν−ε, then we say the primes have level of distribution ν. Thus, according to Theorem 9, the primes are known to have level of distribution ν =1/2. The Elliott-Halberstam conjecture says the primes have level of distribution ν = 1. This remains open. The generalized Mangoldt function is given by k Λk(n)= μ(d) loge (n/d). d|n

∞ Also, let {an}n=1 be a sequence of positive real numbers, −1 A(x)= an, and H = (1 − f(p))(1 − 1/p) , n≤x p for a multiplicative function f. Then the following, proved by Bombieri in 1976—see [3]—under the assumption of the validity of the Elliott-Halberstam conjecture, is called the asymptotic sieve, where k ≥ 2: k−1 anΛk(n) ∼ kHA(x)(loge x) . (7) n≤x

The case k = 2 and an = 1 for all n is essentially Selberg’s sieve. The most striking application to date of (7) was achieved by Friedlander and Iwaniec in 1998—see [4]–[5]—when they proved the following.

Theorem 11 The Friedlander-Iwaniec Theorem There are inﬁnitely many primes of the form a2 + b4.

We have covered an overview of some of the successes of sieve methods, but there are weaknesses. In particular, sieve methods cannot, in general, distinguish between numbers with an even number of prime factors and an odd number of prime factors, which is called the parity problem. Bombieri’s 78 R. A. Mollin sieve clarified some of this issue in [2]–[3], by showing that, assuming the validity of the Elliot-Halberstam conjecture, his sieve implies an asymptotic formula for anF (n) n≤x precisely when a function F provides what is called equal weight to integers with an even number of prime factors and those with an odd number of prime factors. It turns out that the generalized Mangoldt functions have exactly this property for k>1. Of course, the parity problem remains, but the above strides and applications are indicative of the power of sieve methods. It is worth pointing out, before we turn to another topic, that the Elliot- Halberstram conjecture implies some fascinating recent results for gaps between primes as well as implications for the twin-prime conjecture. These were found by Goldston, Pintz, and Yildirim in 2005—see [6]–[7]. For the following statement recall that the infimum of a set S is the greatest lower bound of S and is denoted inf(S). Also, the limit inferior, denoted by lim inf, is given by lim infn→∞an = lim (infm≥nam) n→∞ for a sequence {an}. The first result is unconditional.

Theorem 12 Unconditional Goldston-Pintz-Yildirim

If pn denotes the n-th prime, then

pn+1 − pn n→∞ ∞ lim inf 2 < . loge pn(loge loge pn)

Also, if {an} is a sequence of natural numbers satisfying that

1/2 2 |{an : n ≤ N}| >C(loge N) (log2 N) for all sufficiently large N, then infinitely many of the differences of two elements of {an} can be expressed as the difference of two primes.

The following is the conditional result.

Theorem 13 The Conditional Goldston-Pintz-Yildirim Theorem If the Elliott-Halberstam conjecture is true, then

lim infn→∞pn+1 − pn ≤ 16. An overview of Sieve methods 79

Remark 4 It is worth noting that, in joint work with S. Graham, Goldston, Pintz, and Yildirim proved that if qn is the n-th natural number with exactly two prime factors, then under the assumption of a generalized Elliot- Halberstram conjecture:

lim infn→∞qn+1 − qn ≤ 6

–see: http://www.math.boun.edu.tr/instructors/yildirim/yildirim.htm.

Acknowledgements: The author’s research is supported by NSERC Canada grant # A8484.

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Received: May, 2009